Iterative refinement from a 24-bit factorisation, κ = 10⁴
At its defaults it draws iterative refinement from a 24-bit factorisation, κ = 10⁴. A semi-logarithmic plot of forward error against refinement step. One curve falls steeply to the level of a double-precision solve; the other is nearly flat.
refinement-ladder is one function in lib/figures/mixed.js —
mixed precision — refinement, and the threshold at 1/u. Everything below came out of it during this build, at
arguments taken from the essays rather than invented for this page. A figure here is the
figure a reader meets in an essay, and if the generator changes, this page changes with it.
At its defaults
Drawn even though every essay passes arguments — which on this site is every essay, at 100% of placements since the standard pass. A default nothing exercises is a trap for the next essay to call this with none, and this is the page where a default that has drifted from the figures around it becomes visible.
A semi-logarithmic plot of forward error against refinement step. One curve falls steeply to the level of a double-precision solve; the other is nearly flat.
bits: 24
The arguments are the ones A condition number scaling cannot move passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
A semi-logarithmic plot of forward error against refinement step. One curve falls steeply to the level of a double-precision solve; the other is nearly flat.
bits: 10
The arguments are the ones Buying the accuracy back passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
A semi-logarithmic plot of forward error against refinement step. One curve falls steeply to the level of a double-precision solve; the other is nearly flat.
bits: 14
The arguments are the ones Buying the accuracy back passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
A semi-logarithmic plot of forward error against refinement step. One curve falls steeply to the level of a double-precision solve; the other is nearly flat.
bits: 16
The arguments are the ones Buying the accuracy back passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
A semi-logarithmic plot of forward error against refinement step. One curve falls steeply to the level of a double-precision solve; the other is nearly flat.
bits: 40
The arguments are the ones Buying the accuracy back passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
A semi-logarithmic plot of forward error against refinement step. One curve falls steeply to the level of a double-precision solve; the other is nearly flat.
What it checked while drawing
Every figure above checked its own claims on the way to being drawn, and a claim that failed
would have stopped the picture rather than shipped a wrong one. Those checks used to leave
no trace at all: a passing one returned true and the only evidence the figure had
checked anything was that nothing crashed. The list below is what they actually said, collected
by running this generator with an observer installed — not a description of
what it is believed to check.
8 distinct claims across 6 sets of arguments, grouped below by shape — because most of them are one sentence with a different number in it, and how many separate times that sentence was put to the test is the informative part.
a double residual recovers the accuracy
and a same-precision residual does not
both runs start from the same solve
LU is for square matrices
matmul shapes agree
past the threshold, refinement does not reach the working precision
reaching what a double solve reaches
the constructed matrix has κ = 10000
Against the rule
It draws a decomposition and prints its residual. It calls
luFactor, refine,
and every figure above carries the badge — which residualcheck verifies by looking
for it in the emitted SVG rather than by finding the call that builds one. A badge that is
constructed and then left out of the body is the failure that check exists for.
Across the library: the rule bites on 217
of 397 generators —
199 print a residual and
18 are exempt with a published reason;
180 factorise nothing.
Read from lib/residual-rule.js, which is the same body the gate enforces from,
and the gate's last check fails the build if this page and it disagree about any generator.
Where it is called
Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.
A condition number scaling cannot move
Skeel's componentwise condition number is invariant under any row scaling — exactly, before any norm is taken, because two diagonal factors cancel entry by entry. It is never larger than the normwise one and can be arbitrarily smaller, and the ratio between them is a diagnostic for which kind of ill-conditioning a matrix has.
The arithmetic underneathBuying the accuracy back
Factorise in single precision, then correct the answer using residuals computed in double, and the result is what a full double-precision solve would have given. Compute those residuals in single instead and the identical algorithm, at identical cost, recovers nothing.